An Efficient Algorithm for Dynamic Pricing Using a Graphical Representation

We study a multi-period, multi-item dynamic pricing problem faced by a retailer. The objective is to maximize the total profit by choosing optimal prices while satisfying several important practical business rules. The strength of our work lies in a graphical model reformulation we introduce, which can be used to solve the problem, while providing access to a whole range of ideas from combinatorial optimization. Contrasting to previous literature, we do not make any assumptions on the structure of the demand functions. The complexity of our method depends linearly on the number of time periods but is exponential in the memory of the model (number of past prices that affect the current demand) and in the number of items. Consequently for problems with large memory, we show that the profit maximization problem is NP-hard by presenting a reduction from the Traveling Salesman Problem. We then approximate general demand functions using the commonly used reference price model that accounts for an exponentially smoothed contribution of all the past prices. For the reference price model, we develop a (1 $\epsilon$)-approximation with low runtimes. We extend the reference price model to handle cross-item effects among multiple items using the notion of a virtual reference price. To allow for scalability of our approach, we cluster the items into blocks, and show how to adapt our methods to incorporate global business constraints. Finally, we apply our solution approaches using demand models calibrated by supermarket data, and show that we can solve realistic size instances in a few minutes.